Consider the eigenvalue problem v = v in the unit square D
Chapter 10, Problem 4(choose chapter or problem)
Consider the eigenvalue problem v = v in the unit square D = {0 < x < 1,0 < y < 1}with the Dirichlet BC v =0 on the bottom and both vertical sides, and the Robin BC v/y = v on the top{y=1}.(a) Show that all the eigenvalues are positive. (b) Find an equation for the eigenvalues . Show that they can be expressed in terms of the roots of the equation s+tans =0. (c) Find the solutions of the last equation graphically. Find an approximate formula for the (m, n)th eigenvalue for large (m, n).
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